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The divergence of a magnetic field is zero.

$$\vec{\nabla}.\vec{B}=0$$

i.e. $$\vec{B}=\vec{\nabla} \times \vec{A}=\vec{\nabla} \times (\vec{A'}+\vec{\nabla}f)$$

Here:

$\vec{B}$ is magnetic field.

$\vec{A}$ is ambiguous magnetic vector potential.

$f$ is arbitrary function of $(x,y,z)$. It indeed is also ambiguous.

What is $\vec{A'}$ called? Is it also ambiguous?

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  • $\begingroup$ Do you hear the rusty gears creak as I answer? Anyway, if you have an $A'$ that has zero curl, then you can add an arbitrary value to $A$ and get the same $B$. Did your calculus tell you something with zero curl? $\endgroup$ – user93146 Jun 29 '18 at 16:49
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    $\begingroup$ The magnetic potential itself is a meaningless just as the electrostatic potential. They both are ambiguous, if you like. $\endgroup$ – Vladimir Kalitvianski Jun 29 '18 at 17:12
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    $\begingroup$ $\vec{A}'$ is the vector potential in a different gauge. $\endgroup$ – probably_someone Jun 29 '18 at 19:02
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I am not sure "ambiguous magnetic vector potential" is actually a technical term (a quick Google search brings up no exact results).

In the literature it is common to call both $\vec A $ and $\vec A'$ magnetic vector potential (or Gauge potentials). Either one is as valid as the other and one as such does not have a different name from any other.

That said, it is common to apply a "Gauge condition" to $\vec A$ such as: $$\vec \nabla \cdot \vec A=0$$ in which case we would call $\vec A$ for example the "magnetic vector potential in the Coulomb gauge".

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  • $\begingroup$ You are right. In electrical engineering we define it as we want, to make "life easier". It doesn't matter if we use Coulomb gauge or Lorenz gauge, but solving a problem (using Maxwell's equation) makes it much easier using the "right gauge". $\endgroup$ – abu_bua Jun 29 '18 at 22:30
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The commonly accepted point of view is that any theory involving electromagnetism should be gauge invariant, that is, invariant under the substitution $\phi, \vec A \rightarrow \phi - \partial \chi / \partial t, \vec A + \vec \nabla \chi$ where $\chi$ is arbitrary. However there are arguments for a critical position. The Aharonov-Bohm effect implies that $\vec A$ is a physical quantity. Classical mechanics, quantum mechanics and quantum electrodynamics cannot be formulated in terms of $\vec E $ and $\vec B$, but require $V$ and $\vec A$. Only in the Lorenz gauge the wave equations for $\phi, \vec A$ are compliant with causality. There are numerous paradoxes involving the electromagnetic conservation laws. In my paper I argue that an alternative formulation is possible that resolves the paradoxes and respects causality. It states the relation known as the Lorenz gauge as the consequence of charge-current conservation.

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